Class Numbers of Orders in Quartic Fields Dissertation der FakultÄat furÄ Mathematik und Physik der Eberhard-Karls-UniversitÄat TubingenÄ zur Erlangung des Grades eines Doktors der Naturwissenschaften vorgelegt von Mark Pavey aus Cheltenham Spa, UK 2006 Mundliche Prufung:Ä 11.05.2006 Dekan: Prof.Dr.P.Schmid 1.Berichterstatter: Prof.Dr.A.Deitmar 2.Berichterstatter: Prof.Dr.J.Hausen Dedicated to my parents: Deryk and Frances Pavey, who have always supported me. Acknowledgments First of all thanks must go to my doctoral supervisor Professor Anton Deit- mar, without whose enthusiasm and unfailing generosity with his time and expertise this project could not have been completed. The ¯rst two years of research for this project were undertaken at Exeter University, UK, with funding from the Engineering and Physical Sciences Re- search Council of Great Britain (EPSRC). I would like to thank the members of the Exeter Maths Department and EPSRC for their support. Likewise I thank the members of the Mathematisches Institut der UniversitÄat Tubingen,Ä where the work was completed. Finally, I wish to thank all the family and friends whose friendship and encouragement have been invaluable to me over the last four years. I have to mention particularly the Palmers, the Haywards and my brother Phill in Seaton, and those who studied and drank alongside me: Jon, Pete, Dave, Ralph, Thomas and the rest. Special thanks to Paul Smith for the Online Chess Club, without which the whole thing might have got done more quickly, but it wouldn't have been as much fun. 5 Contents Introduction iii 1 Euler Characteristics and In¯nitesimal Characters 1 1.1 The unitary duals of SL2(R) and SL4(R) . 1 1.2 Euler characteristics . 5 1.3 Euler-Poincar¶e functions . 9 1.4 Euler characteristics in the case of SL4(R) . 15 1.5 The unitary dual of KM . 19 1.6 In¯nitesimal characters . 21 2 Analysis of the Ruelle Zeta Function 28 2.1 The Selberg trace formula . 30 2.2 The Selberg zeta function . 32 2.3 A functional equation for ZP,(s) . 37 2.4 The Ruelle zeta function . 43 2.5 Spectral sequences . 47 2.6 The Hochschild-Serre spectral sequence . 48 2.7 Contribution of the trivial representation . 51 2.8 Contribution of the other representations . 53 3 A Prime Geodesic Theorem for SL4(R) 56 3.1 Analytic properties of R¡,(s) . 57 3.2 Estimating Ã(x) and Ã~(x) . 61 3.3 The Wiener-Ikehara Theorem . 66 3.4 The Dirichlet series . 71 3.5 Estimating Ãn(x) . 76 3.6 Estimating Ã1(x) . 79 3.7 Estimating ¼(x), ¼~(x) and ¼1(x) . 81 i ii 4 Division Algebras of Degree Four 85 4.1 Central simple algebras and orders . 85 4.2 Division algebras of degree four . 87 4.3 Sub¯elds of M(Q) generated by ¡ . 89 4.4 Field and order embeddings . 90 4.5 Counting order embeddings . 96 5 Comparing Geodesics and Orders 108 5.1 Regular geodesics . 108 5.2 Class numbers of orders in totally complex quartic ¯elds . 117 Introduction In this thesis we present two main results. The ¯rst is a Prime Geodesic Theorem for compact symmetric spaces formed as a quotient of the Lie group SL4(R). The second is an application of the Prime Geodesic Theorem to prove an asymptotic formula for class numbers of orders in totally complex quartic ¯elds with no real quadratic sub¯eld. Before stating our results we give some background. Let D be the set of all natural numbers D 0; 1 mod 4 with D not a square. Then D is the set of all discriminants ´of orders in real quadratic ¯elds. For D D the set 2 x + ypD D = : x yD mod 2 O ( 2 ´ ) is an order in the real quadratic ¯eld Q(pD) with discriminant D. As D varies, runs through the set of all orders of real quadratic ¯elds. For OD D D let h( D) denote the class number and R( D) the regulator of the order2 . ItOwas conjectured by Gauss ([22]) andOproved by Siegel ([55]) OD that, as x tends to in¯nity ¼2x3=2 h( )R( ) = + O(x log x); D D 18³(3) D2D O O XD·x where ³ is the Riemann zeta function. For a long time it was believed to be impossible to separate the class number and the regulator in the summation. However, in [51], Theorem 3.1, Sarnak showed, using the Selberg trace formula, that as x we have ! 1 e2x h( ) : D 2x D2D O » R(XOD)·x iii Introduction iv More sharply, 3x=2 2 h( D) = L(2x) + O e x D2D O R(XOD)·x ¡ ¢ as x , where L(x) is the function ! 1 x et L(x) = dt: t Z1 Sarnak established this result by identifying the regulators with lengths of closed geodesics of the modular curve SL2(Z) H, where H denotes the upper half-plane, and using the Prime Geodesic Theoremn for this Riemannian sur- face. Actually Sarnak proved not this result but the analogue where h( ) is replaced by the class number in the narrower sense and R( ) by a \regula-O O tor in the narrower sense". But in Sarnak's proof the group SL2(Z) can be replaced by PGL2(Z) giving the above result. The Prime Geodesic Theorem in this context gives an asymptotic formula for the number of closed geodesics on the surface SL2(Z) H with length less than or equal to x > 0. This formula is analogous to the asymptoticn formula for the number of primes less than x given in the Prime Number Theorem. The Selberg zeta function (see [52]) is used in the proof of the Prime Geodesic Theorem in a way analogous to the way the Riemann zeta function is used in the proof of the Prime Number Thoerem (see [9]). The required properties of the Selberg zeta function are deduced from the Selberg trace formula ([52]). It seems that following Sarnak's result no asymptotic results for class numbers in ¯elds of degree greater than two were proven until in [15], Theo- rem 1.1, Deitmar proved an asymptotic formula for class numbers of orders in complex cubic ¯elds, that is, cubic ¯elds with one real embedding and one pair of complex conjugate embeddings. Deitmar's result can be stated as follows. Let S be a ¯nite set of prime numbers containing at least two elements and let C(S) be the set of all complex cubic ¯elds F such that all primes p S are non-decomposed in F . For F C(S) let OF (S) be the set of all2isomorphism classes of orders in F whic2h are maximal at all p S, ie. 2 are such that the completion = Z is the maximal order of the ¯eld Op O ­ p Fp = F Qp for all p S. Let O(S) be the union of all OF (S), where F ranges ov­er C(S). For a2¯eld F C(S) and an order O (S) de¯ne 2 O 2 F ¸ ( ) = ¸ (F ) = f (F ); S O S p p2S Y Introduction v where fp(F ) is the inertia degree of p in F . Let R( ) denote the regulator and h( ) the class number of the order . O ForOx > 0 we de¯ne O ¼ (x) = ¸ ( )h( ): S S O O O2O(S) RX(O)·x Then as x we have ! 1 e3x ¼ (x) : S » 3x More sharply, e9x=4 ¼ (x) = L(3x) + O S x µ ¶ as x . This! 1result was again proved by means of a Prime Geodesic Theorem, this time for symmetric spaces formed as a compact quotient of the group SL3(R). There exists a Selberg trace formula for such spaces ([59]) by means of which the required properties of a generalised Selberg zeta function can be deduced ([14]) in order to prove the Prime Geodesic Theorem. The class number formula is then deduced by means of a correspondence between primitive closed geodesics and orders in complex cubic number ¯elds, under which the lengths of the geodesics correspond to the regulators of the number ¯elds. In this thesis we follow the methods of [15]. In the ¯rst three chapters we prove a Prime Geodesic Theorem for compact quotients of SL4(R). In the ¯nal two chapters we give an arithmetic interpretation in terms of class numbers. Our main results can be stated as follows. Let G = SL4(R) and let K be the maximal compact subgroup SO(4). Let ¡ G be discrete and cocompact. We then have a one to one correspondence bet½ween conjugacy classes in ¡ and free homotopy classes of closed geodesics on the symmetric space X = ¡ G=K. Let ¡ n a a A¡ = 8 : 0 < a < 19 0 a¡1 1 > > <B a¡1 C = B C >@ A > and :> ;> SO(2) B = : SO(2) µ ¶ Introduction vi Let E (¡) be the set of primitive conjugacy classes [] in ¡ such that is conjugate in G to an element a b of A¡B. For ¡ we write a also 2 for the top left entry in the matrix a and de¯ne the length l of to be 8 log a. Let G; ¡ be the centralisers of in G and ¡ respectively and let K = K G . For x > 0 de¯ne the function \ ¼(x) = Â1(¡); []2E (¡) eXl ·x where Â1(¡) is the ¯rst higher Euler characteristic of the symmetric space X = ¡ G =K . ¡ n Theorem 0.0.1 (Prime Geodesic Theorem) For x we have ! 1 2x ¼(x) : » log x More sharply, x3=4 ¼(x) = 2 li(x) + O log x µ ¶ as x , where li(x) = x 1 dt is the integral logarithm.